The Nonexistence of Ternary [38, 6, 23] Codes

It has been shown by Bogdanova and Boukliev 1] that there exist a ternary 38,5,24] code and a ternary 37,5,23] code. But it is unknown whether or not there exist a ternary 39,6,24] code and a ternary 38,6,23] code. The purpose of this paper is to prove that (1) there is no ternary 39,6,24] code and (2) there is no ternary 38,6,23] code using the nonexistence of ternary 39,6,24] codes. Since it is known (cf. Brouwer and Sloane 2] and Hamada and Watamori 14]) that (i) n₃(6,23) = 38> or 39 and d₃(38,6) = 22 or 23 and (ii) n₃(6,24) = 39 or 40 and d₃(39,6) = 23 or 24, this implies that n₃(6,23) = 39, d₃(38,6) = 22, n₃(6,24) = 40 and d₃(39,6) = 23, where n3<>(k,d) and d<>3(n,k) denote the smallest value of n and the largest value of d, respectively, for which there exists an n,k,d] code over the Galois field GF(3).